exp-w (used to crash)

Average Error: 0.3 → 0.3

Time: 59.2s

Precision: binary64

Cost: 19456

Math

\[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

↓

\[\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

↓

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (exp w)))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

↓

double code(double w, double l) {
	return pow(l, exp(w)) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

↓

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l ** exp(w)) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

↓

public static double code(double w, double l) {
	return Math.pow(l, Math.exp(w)) / Math.exp(w);
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

↓

def code(w, l):
	return math.pow(l, math.exp(w)) / math.exp(w)

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

↓

function code(w, l)
	return Float64((l ^ exp(w)) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

↓

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.3

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   Proof

   [Start] 0.3	\[ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   rational_best-simplify-1 [=>] 0.3	\[ \color{blue}{{\ell}^{\left(e^{w}\right)} \cdot e^{-w}} \]
   exponential-simplify-2 [=>] 0.3	\[ {\ell}^{\left(e^{w}\right)} \cdot \color{blue}{\frac{1}{e^{w}}} \]
   rational_best-simplify-55 [=>] 0.3	\[ \color{blue}{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
   rational_best-simplify-1 [=>] 0.3	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot 1} \]
   rational_best-simplify-7 [=>] 0.3	\[ \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

3. Final simplification 0.3

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]

Alternatives

Alternative 1
Error	1.5
Cost	13376

\[\left(1 + w \cdot \log \ell\right) \cdot \frac{\ell}{e^{w}} \]

Alternative 2
Error	1.3
Cost	13376

\[\frac{\ell + \log \ell \cdot \left(\ell \cdot w\right)}{e^{w}} \]

Alternative 3
Error	1.9
Cost	6592

\[\frac{\ell}{e^{w}} \]

Alternative 4
Error	2.4
Cost	1344

\[\begin{array}{l} t_0 := w + \left(w + -2\right)\\ \frac{t_0 \cdot \left(w \cdot \left(\ell \cdot -2\right) + \ell \cdot t_0\right)}{4} \end{array} \]

Alternative 5
Error	8.7
Cost	1284

\[\begin{array}{l} \mathbf{if}\;w \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\left(2 - \left(w + 1\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\ell - \left(\ell \cdot \left(w \cdot 1.5\right) + -1\right)\right) + \left(-\left(1 + \ell \cdot \left(w \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	8.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;w \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\left(2 - \left(w + 1\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 2 \cdot \left(\ell \cdot w\right)\right) + \left(\ell - \left(1 - \ell \cdot w\right)\right)\\ \end{array} \]

Alternative 7
Error	8.7
Cost	772

\[\begin{array}{l} \mathbf{if}\;w \leq 0.096:\\ \;\;\;\;\left(2 - \left(w + 1\right)\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \ell \cdot w\right) + \left(1 - \left(-\ell\right)\right)\\ \end{array} \]

Alternative 8
Error	14.1
Cost	64

\[\ell \]

Reproduce

herbie shell --seed 2023099 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))